Publ. Astron. Soc. Japan 2014 66 (4), L6 (1–5)
doi: 10.1093/pasj/psu049
L6-1
Advance Access Publication Date: 2014 July 14
Letter
Letter
Scaling laws of free magnetic energy stored in
a solar emerging flux region
Tetsuya MAGARA∗
Department of Astronomy and Space Science, School of Space Research, Kyung Hee University, 1732
Deogyeong-daero, Giheung-gu, Yongin, Gyeonggi-do, 446-701, Republic of Korea
*E-mail: [email protected]
Received 2014 April 9; Accepted 2014 May 7
Abstract
This Letter reports scaling laws of free magnetic energy stored in a solar emerging flux
region which is a key to understanding the energetics of solar active phenomena such
as solar flares and coronal mass ejections. By performing three-dimensional magnetohydrodynamic simulations that reproduce several emerging flux regions of different
magnetic configurations, we derive power-law relationships among emerged magnetic
flux, free magnetic energy and relative magnetic helicity in these emerging flux regions.
Since magnetic flux is an observable quantity, the scaling law between magnetic flux
and free magnetic energy may give a way to estimate invisible free magnetic energy
responsible for solar active phenomena.
Key words: magnetohydrodynamics: MHD — Sun: activity — Sun: coronal mass ejections (CMEs) — Sun: flares —
Sun: magnetic fields
1 Introduction
While the Sun usually appears to be quiet in white light, the
Sun is actually full of active phenomena such as solar flares
(Shibata & Magara 2011) and coronal mass ejections (Chen
2011); they are now observed on a daily basis. These active
phenomena often occur in an emerging flux region with
intense magnetic flux in it, which is called an active region.
Since they are observed as one of the largest energy-release
events on the Sun, the energetics of those active phenomena
has been an important target of research in solar physics.
Some of them even affect the environment of the Earth;
in an extreme case they damage telecommunication and
power supplies. This makes it important to investigate the
energetics of solar active phenomena and their impact on
the Earth, which has emerged as a main target of space
weather.
When we study the energetics of solar active phenomena,
it is important to estimate the free magnetic energy required
to produce an energy-release event. The free energy is
defined as an excess over the potential energy that is the
lowest energy taken by a magnetic structure formed on the
Sun. Since it is still difficult to derive the overall configuration of a magnetic structure only by observations, especially
in the solar corona where most of huge active phenomena
originate, we cannot directly calculate free magnetic energy;
instead we have to use observable quantities to estimate free
magnetic energy.
One of the ways to estimate free magnetic energy is to
reconstruct a coronal magnetic structure from an observed
photospheric magnetic field. This has become useful when
not only the vertical component of a photospheric field
but also its transverse components are available, so that by
C The Author 2014. Published by Oxford University Press on behalf of the Astronomical Society of Japan.
All rights reserved. For Permissions, please email: [email protected]
L6-2
Publications of the Astronomical Society of Japan, 2014, Vol. 66, No. 4
combining these components we can reconstruct a magnetic
structure containing free magnetic energy in the corona.
Reconstruction of a coronal magnetic structure based on
the force-free approximation (e.g., Sturrock 1994) has well
been investigated and developed (Wiegelmann & Sakurai
2012), and it has widely been applied to observed active
regions to derive their magnetic configurations. There have
been proposed various nonlinear force-free field (NLFFF)
reconstruction methods and reviews of them with a
quantitative comparison are given in Schrijver et al. (2006)
and Metcalf et al. (2008). Some recent work where NLFFF
reconstruction methods are used for investigating observed
active regions is found in Metcalf, Leka, and Mickey(2005),
Régnier and Priest (2007), Inoue et al. (2012), and Sun
et al. (2012), the last of which use an optimization code
developed by Wiegelmann (2004). The validity of NLFFF
reconstruction and how to improve it have also been studied
(Metcalf et al. 1995; Wheatland & Metcalf 2006).
In this Letter we report scaling laws of free magnetic
energy stored in an emerging flux region, which may give
a supplementary way to estimate invisible free magnetic
energy using observed magnetic flux. By performing a
series of three-dimensional magnetohydrodynamic (MHD)
simulations that reproduce several emerging flux regions
by bringing a magnetic flux tube of different twist to
the solar atmosphere, we derive power-law relationships
among free magnetic energy, magnetic flux and relative
magnetic helicity, the last of which is also an important quantity related to the activity of an emerging flux
region (Jeong & Chae 2007; Lim et al. 2007). We also
use a linear force-free field (LFFF) model to investigate
power-law relationships among these quantities. By comparing the simulations and the LFFF model we discuss the
possible application of the scaling laws to an emerging
flux region where a huge active phenomenon tends
to occur.
2 Simulation set-up and results
To perform a three-dimensional simulation we solved ideal
MHD equations, the details of which are described in
Magara (2012). We use the Cartesian coordinates (x, y, z)
where the x- and y-axes form a horizontal plane corresponding to the solar surface (z = 0), while the z-axis is
directed upward. The present study uses a wider simulation
domain (−200, −200, −10) < (x, y, z) < (200, 200, 190)
than that used in Magara (2012) so that long-term evolution can be investigated. The domain is discretized into grids
whose size is (x, y, z) = (0.1, 0.2, 0.1) for (−8, −12,
−10) < (x, y, z) < (8, 12, 15) and it gradually increases
up to (4, 4, 4) as |x|, |y|, and z increase. The total number
of grids is Nx × Ny × Nz = 371 × 303 × 353. Inside this
domain we put a background atmosphere stratified under
uniform gravity. A magnetic flux tube with a finite radius
rf = 2, characterized by a Gold–Hoyle profile (Gold &
Hoyle 1960) initially submerges under the solar surface.
The axis of this flux tube is parallel to the y-axis, crossing
the z-axis at z = −4. The flux tube and background atmosphere are initially in mechanical equilibrium, and to initiate
a simulation we applied a velocity perturbation to the
middle portion of the flux tube [we followed equation (10)
of Magara (2012), except for using L = 200 in the present
study], which has then started to emerge in the shape of an
loop. The unit of normalization is given by 2ph (length),
2
(gas pressure),
cph (velocity), ρ ph (gas density), ρph cph
2 1/2
Tph (temperature), and (ρph cph
)
(magnetic field) where
ph , cph , ρ ph , and Tph represent the pressure scale height,
adiabatic sound speed, gas density, and temperature in the
photosphere, respectively.
We ran a series of simulations by changing the twist
parameter of field lines composing a Gold–Hoyle flux
tube. In Magara (2012) we studied four cases where the
twist parameter is given by 0.2, 0.35, 0.5, and 1; in the
present study we add another case (twist parameter is 0.8).
Figures 1a–1c present snapshots of an emerging flux region
in this newly added case, taken at t = 31 from a perspective
(figure 1a), top (figure 1b), and side viewpoint (figure 1c).
In these figures field lines emerging below the surface (greyscale map) are displayed in color. Figures 1d–1f show the
time variations of emerged magnetic flux, free magnetic
energy and relative magnetic helicity stored in the atmosphere z ≥ 0, which are obtained by following the same
procedure as described in Magara and Longcope (2003)
and Magara (2012). The red vertical lines in these figures
indicate t = 31.
We also derived these time-series plots in all the other
cases. We have then investigated mutual relationships
among emerged magnetic flux, free magnetic energy, and
relative magnetic helicity during the dynamic evolution of
an emerging magnetic field. Figure 2a shows a graph of
emerged flux vs. free magnetic energy in a logarithmic scale,
while in figures 2b and 2c log–log plots of emerged flux vs.
relative magnetic helicity and relative magnetic helicity vs.
free magnetic energy are presented. In these figures the red,
blue, green, orange and violet lines represent that the twist
parameter is given by 0.2, 0.35, 0.5, 0.8, and 1, respectively. A black dashed line in each figure indicates that the
power-law index is given by 2 (figure 2a), 2 (figure 2b), and
1 (figure 2c). These indices are derived from the following
dimensional analysis: ∼ BL2 ∝ B, Em ∼ B2 L3 ∝ B2 , and
Hm ∼ B2 L4 ∝ B2 where , Em and Hm represent emerged
flux, free magnetic energy and relative magnetic helicity,
while B and L are magnetic field strength and length,
respectively.
Publications of the Astronomical Society of Japan, 2014, Vol. 66, No. 4
L6-3
Fig. 1. (a)–(c) Snapshots of an emerging magnetic field taken at t = 31 from a perspective (a), top (b), and side (c) viewpoint are presented. The twist
parameter is 0.8. Field lines are represented by colored lines and photospheric magnetic flux is given by a grey-scale map (white region has positive
polarity while black region has negative polarity) at z = 0. (d)–(f) Time variations of emerged magnetic flux (d), free magnetic energy (e) and relative
magnetic helicity (f) are presented. The red vertical lines indicate t = 31. (Color online)
Fig. 2. (a) log–log plot of emerged magnetic flux vs. free magnetic energy is presented. The twist parameter is 0.2 (red), 0.35 (blue), 0.5 (green),
0.8 (orange), and 1 (violet). The dashed line indicates power-law index = 2. (b) Same as (a), except for a plot of emerged magnetic flux vs. relative
magnetic helicity. The dashed line indicates power-law index = 2. (c) Same as (a), except for a plot of relative magnetic helicity vs. free magnetic
energy. The dashed line indicates power-law index = 1. (Color online)
3 Discussion
Figures 2a–2c show characteristics of emerged magnetic
flux, relative magnetic helicity and free magnetic energy.
In an ideal MHD system the first two quantities are conserved quantities, that is, as long as they are injected into
the solar atmosphere through the emergence of a twisted
flux tube, both of these quantities continuously increase
there. On the other hand, during the dynamic evolution of
an emerging magnetic field, free magnetic energy is continuously converted into the kinetic energy of expansion
of an emerging field. This may be one of the reasons that
a deviation from a power-law fitting line is observed in the
plots of free magnetic energy vs. one of these conserved
quantities.
L6-4
Publications of the Astronomical Society of Japan, 2014, Vol. 66, No. 4
Fig. 3. (a) Schematic illustration of a 2.5D linear force-free field given by equations (1)– (3) is presented. The y-axis represents a PIL while γ is a shear
vs. E m given by equations (5) and (6) is represented by a black line. The red and blue dashed lines show asymptotic
angle. (b) A log–log plot of Hm
lines given by equations (7) and (8), respectively. Here α = 0.5. (Color online)
In this respect, it might be possible to have a situation
where the amount of free magnetic energy converted to
kinetic energy is significantly small compared to the amount
stored in an emerging flux region, especially in the core of
an emerging flux region where a magnetic field shows quasistatic behavior (Magara & Longcope 2003). Let us consider
a simple model of an emerging flux region comprised of a
static field where no free magnetic energy is converted. We
use an LFFF given by (e.g., Priest 1982)
Bx = − 1 −
α 2
k
B0 cos (kx) e
√
α
2
2
By = − B0 cos (kx) e− k −α z ,
k
√
− k2 −α 2 z
,
Bz = B0 sin (kx) e
=
Hm =
(1)
(2)
α
,
√
k2 k2 − α 2
k4
(3)
where k is the wavenumber while α and B0 are constants
representing the force-free parameter and field strength.
An emerging flux region typically has the following evolutionary features (e.g., Shibata & Magara 2011 and references therein). (1) At the beginning of emergence, the area of
emerged flux at the surface is small and it gradually extends
along a polarity inversion line (PIL) as emergence goes on.
The direction of the PIL basically corresponds to the axis of
an emerging flux tube. (2) At the beginning of emergence,
the direction of an emerging field is almost perpendicular
to a PIL, while it is gradually aligned with the PIL to form
a sheared loop. To incorporate these basic features into the
model, we assume a configuration illustrated in figure 3a.
(4)
α 2
,
− α 2
(5)
k2
and
E m =
,
and
√
− k2 −α 2 z
By changing k from ∞ toward α, we make pseudo-evolution
of an emerging flux region, although the whole magnetic
field is always in a static state. We then calculate emerged
flux, relative magnetic helicity and free magnetic energy
, 0, 0) ≤ (x, y, z) ≤ ( 2π
, πk √k2α−α2 , ∞),
in the domain (− 2π
k
k
given by (e.g., Berger 1985)
√
k2 − α 2 α ,
k3 k2 − α 2
k −
(6)
respectively. Here means that a quantity is given in a
dimensionless form. From equations (4) and (5) we can
immediately obtain Hm = 2 , which is consistent with the
simulations. A relation between E m and Hm (or ) is a
bit complicated; in the following we present an asymptotic
result on this relation when k → ∞ and k → α .
E m ∼
1 1
α Hm = α 2
2
2
E m ∼ α Hm = α 2
for k → ∞
for k → α (7)
(8)
In figure 3b the black line shows a log–log plot of Hm vs.
E m while red and blue dashed lines represent equations (7)
and (8), respectively. This suggests that power-law indices
Publications of the Astronomical Society of Japan, 2014, Vol. 66, No. 4
for the ( , E m)-plot and the (Hm , E m)-plot are given by
2 and 1, which are the same as shown in figures 2a and 2c.
Comparing the simulations and the LFFF model raises
a question about how much free magnetic energy remains
in a magnetic structure emerging and evolving toward a
well-developed state ready for producing an energy-release
event. This question becomes more important when we use
the scaling laws to estimate the remaining free magnetic
energy in an emerging flux region. To answer this question
we need wide and detailed investigations of how the energy
conversion of free magnetic energy proceeds in emerging
flux regions of various magnetic configurations and how
well force-free models give an estimate of free magnetic
energy stored in these regions. For this purpose we will
try to use a more general force-free field model than the
LFFF model presented here and compare it to the simulations. Results of a detailed comparison will be reported in
a forthcoming paper.
Acknowledgement
The author wishes to thank the Kyung Hee University for general support of this study. He also appreciates useful comments
given by the referee. This study was financially supported by Basic
Science Research Program (NRF-2013R1A1A2058705, PI: T.
Magara) through the National Research Foundation of Korea (NRF)
provided by the Ministry of Education, Science and Technology, as
well as by the BK21 plus program through the NRF.
L6-5
References
Berger, M. A. 1985, ApJS, 59, 433
Chen, P. F. 2011, Living Rev. Sol. Phys., 8, 1
Gold, T., & Hoyle, F. 1960, MNRAS, 120, 89
Inoue, S., Shiota, D., Yamamoto, T. T., Pandey, V. S., Magara, T.,
& Choe, G. S. 2012, ApJ, 760, 17
Jeong, H., & Chae, J. 2007, ApJ, 671, 1022
Lim, E.-K., Jeong, H., & Chae, J. 2007, in ASP Conf. Ser., 369,
Proc. New Solar Physics with Solar-B Mission, ed. K. Shibata
et al. (San Francisco: ASP), 175
Magara, T. 2012, ApJ, 748, 53
Magara, T., & Longcope, D. W. 2003, ApJ, 586, 630
Metcalf, T. R., et al. 2008, Sol. Phys., 247, 269
Metcalf, T. R., Jiao, L., McClymont, A. N., Canfield, R. C., &
Uitenbroek, H. 1995, ApJ, 439, 474
Metcalf, T. R., Leka, K. D., & Mickey, D. L. 2005, ApJ, 623,
L53
Priest, E. R. 1982, Solar Magnetohydrodynamics (Dordrecht: D.
Reidel), 74
Régnier, S., & Priest, E. R. 2007, ApJ, 669, L53
Schrijver, C. J., et al. 2006, Sol. Phys., 235, 161
Shibata, K., & Magara, T. 2011, Liv. Rev. Sol. Phys., 8, 6
Sturrock, P. A. 1994, Plasma Physics, An Introduction to the
Theory of Astrophysical, Geophysical and Laboratory Plasmas
(Cambridge: Cambridge University Press)
Sun, X., Hoeksema, J. T., Liu, Y., Chen, Q., & Hayashi, K. 2012,
ApJ, 757, 149
Wheatland, M. S., & Metcalf, T. R. 2006, ApJ, 636, 1151
Wiegelmann, T. 2004, Sol. Phys., 219, 87
Wiegelmann, T., & Sakurai, T. 2012, Liv. Rev. Sol. Phys., 9, 5